Generalized rough function with one variable and its properties

Estimation of uncertainty in knowledge extracted from a repository of data is a challenging issue in artificial intelligence and cognitive sciences. The rough set theory is a prominent technique for data mining from a repository of domain data. A rough set is a formal approximation of a crisp set in terms of a pair of sets with respect to equivalence relations. The concept of lower and upper approximations is based on the assumption that information is associated with every object described by a set of attributes. The classical rough set theory with the equivalence class concept and static dependency cannot handle any uncertainty in attribute level which seems to be a major limitation. Several extensions on classical rough set theory have remarkable contributions in different applications. But excavation of the nature of dependency of the attributes and the estimation of uncertainty is still an unsolved issue. This paper proposes an extension in the classical rough set theory by introducing probabilistic relevancy of the attributes. Mining the correlation between attributes in terms of the probabilistic dependency coefficient is an important way out in discovering the uncertainty. The roughness in contributing knowledge by an attribute can be represented using approximations in the probabilistic dependency space. Using the approximations in the values of the probabilistic dependency coefficient, the inherent uncertainty can be estimated and handled more precisely.

Local rough set: A solution to rough data analysis in big data

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

The definition of basic rough sets [3] depends upon either a single equivalence relation defined on a universe or several equivalence relations defined over the universe, taken one each taken at a time. In the view of granular computing, classical rough set theory is based upon single granulation. Extending this notion, a rough set model based on multi-granulations (MGRS) was introduced in [5]. In this approach, approximations of sets were defined through multiple equivalence relations on the universe and their properties were investigated. Using hybridization of fuzzy set [13] with rough set the concept of rough fuzzy set was introduced by Dubois and Prade [1]. Recently, a Rough Fuzzy Set Model was introduced and studied by Wu and Kou [12], which is based on Multiple Granulation. Topological properties of rough sets introduced by Pawlak in terms of their types were recently studied by Tripathy and Mitra [10]. These were extended to the context of incomplete multi granulation by Tripathy and Raghavan [11]. In this paper we introduce incomplete multigranulation on rough fuzzy sets, study their basic properties and extend the topological properties in [11] to this context. Our findings are true for both complete and incomplete fuzzy rough set models based upon multi granulation.KeywordsRough SetsFuzzy rough setsequivalence relationstolerance relationstype of rough setsmulti granular fuzzy rough sets

Characteristic numbers and approximation operators in generalized rough approximation system

This paper presents an approach to approximating sensor signals. In classical rough set theory, set approximation is carried out in non-empty, finite universes of objects. In contrast, we carry out set approximation inside non-empty, uncountable sets (universes) of points. This study is motivated by an interest in classifying sample values for various types of sensors. The result of this study is the introduction of a family of discrete rough integrals based on rough set theory. The discrete rough integrals have practical implications, since these integrals serve as an aid in approximate reasoning and in pattern recognition relative to segments of continuous signals. In the context of approximate reasoning, discrete rough integrals provide a basis for determining the relevance of sensors over a particular sampling period. In the context of pattern recognition, discrete rough integrals can be useful in doing such things as classifying radar weather data, vehicular traffic patterns, and waveforms of power system faults. By way of illustration, one form of discrete integral is used to assess the accuracy of set approximation of a sensor signal.

A vague-rough set approach for uncertain knowledge acquisition

Rough set theory for the interval-valued fuzzy information systems

Classical rough set theory (RST) can't process incomplete information system (IIS) because it is based on an indiscernibility relation which is a kind of equivalent relation. In the literature a non-symmetric similarity relation based rough set model (NS-RSM) has been introduced as an extended model under IIS with ``?" values directly. Unfortunately, in this model objects in the same similarity class are not necessarily similar to each other and may belong to different target classes. In this paper, a new inequivalent relation called Maximal Limited Consistent block relation (MLC) is proposed. The proposed MLC relation improves the lower approximation accuracy by finding the maximal limited blocks of indiscernible objects in IIS with ``?" values. Maximal Limited Similarity rough set model (MLS) is introduced as an integration between our proposed relation (MLC) and NS-RSM. The resulted MLS model works efficiently under IIS with ``?" values. Finally, an illustrative example is given to validate MLS model. Furthermore, approximation accuracy comparisons have been conducted among NS-RSM and MLS. The results of this work demonstrate that the MLS model outperform NS-RSM.

Variable Precision Rough Set (VPRS) model is one of the most important extensions of the Classical Rough Set (RS) theory. It employs a majority inclusion relation mechanism in order to make the Classical RS model become more fault tolerant, and therefore the generalization of the model is improved. This paper can be viewed as an extension of previous investigations on attribution reduction problem in VPRS model. In our investigation, we illustrated with examples that the previously proposed reduct definitions may spoil the hidden classification ability of a knowledge system by ignoring certian essential attributes in some circumstances. Consequently, by proposing a new ß-consistent notion, we analyze the relationship between the structures of Decision Table (DT) and different definitions of reduct in VPRS model. Then we give a new notion of /3-complement reduct that can avoid the defects of reduct notions defined in previous literatures. We also supply the method to obtain the /3- complement reduct using a decision table splitting algorithm, and finally demonstrate the feasibility of our approach with sample instances.

Decision-making in information systems often involves uncertainty and imprecision. Traditional methods, such as those based on classical Rough Set Theory and original Pawlak’s model may struggle to handle such complexities and various data set class types. Keeping in mind the importance of similarity or dissimilarity measures and applications in data mining, medical diagnosis, decision making, and pattern recognition. This study proposes a novel approach to estimative similarity or dissimilarity degree membership calculation with fuzzy decision-making systems, leveraging symmetry relationships. Our method aims to enhance decision-making accuracy and robustness by considering the inherent uncertainties present in real-world data. Experimental results for selected application of dataset represents a hypothetical medical diagnosis scenario demonstrate the superiority of our approach compared to existing techniques, making it a promising tool for various applications in information systems. By calculating estimative membership degrees based on these relationships and similarity or dissimilarity, our approach offers a more nuanced understanding of decision boundaries and uncertainties.

The classical rough set theory based on complete information systems stems from the observation that objects with the same characteristics are indiscernible according to available information. With respect to upper-approximation and lower-approximation defined on an indiscernibility relation it classifies objects into different equivalent classes. But in some cases such a rigid indiscernibility relation is far from applications in the real world. Therefore, several generalizations of the rough set theory have been proposed some of which extend the indiscernibility relation using more general similarity or tolerance relations. For example, Kryszkiewicz [4] studied a tolerance relation, and Stefanowski [7] explored a non-symmetric, similarity relation and valued tolerance relation. Unfortunately, All the extensions mentioned above have their inherent limitations. In this paper, after discussing several extension models based on rough sets for incomplete information, a concept of constrained dissymmetrical similarity relation is introduced as a new extension of the rough set theory, the upper-approximation and the lower-approximation defined on constrained similarity relation are proposed as well. Furthermore, we present the comparison between the performance of these extended relations. Analysis of results shows that this relation works effectively in incomplete information and generates rational object classification

Covering generalized rough set theory is an important extension of classical rough set theory. To characterize a fuzzy set in a given covering approximation space, a pair of fuzzy sets, called covering rough fuzzy lower and upper approximations, were introduced, but they do not describe well how much uncertainty is induced by the granularity of knowledge. In this paper, we first discuss the relationship between uncertainty and granularity of knowledge. Then we examine several commonly used distance measures, and indicate that some of them exhibit some limitations. Next we propose a roughness measure based on Minkowski distance, and examine some important properties of this measure. Finally, an illustrative example is provided to demonstrate the application of the roughness measure to incomplete information systems with fuzzy decision.

Dominance-based Rough Set Approach (DRSA) is an extension of classical Rough Set Theory (RST) to deal with inconsistent problems. Approximation of class unions is the key method of DRSA and it is a very complex computing job. However, the formal work within DRSA only proposed relative definitions of the concept including approximation of class unions, but no efficient algorithm has been given. So we can only use the original definition to get the approximation of class unions. As a result, it is very complex, and it requires a considerable amount of computing time. In this paper a dominance-matrix is developed and a new method is presented for the approximation of class unions based on dominance-matrix. It simplifies the calculation process and decreases the computing time.

The classical rough set theory is not useful for analyzing incomplete information system,and this problem is solved by tolerance relation,similarity relation,limited tolerance relation and completed tolerance relation to some extent.Limitations of these models were analyzed,and on the basis of the completed tolerance relation an extended rough set model based on improved complete tolerance relation was developed.The extended model not only retains the merits of the existing models,but also overcomes the limitations of the existing models to some extent.The result of case studies shows the effectiveness of the presented model.